PROBABlLlTY
AND
MATHEMATICAL STATISTICS
VoL 14, Fasc 2 (1993). pp 281-285
COMPARISON OF SUMS OF INDEPENDENT
LT3ENTICALLY DISTNBUmD RANDOM VECTORS
S. J. MONTGOMERY-SMITH (COLUMB~A,
MISSOURI)
Abstract. Let S, be the k-th partial sum of Banach space valued
independent identically distributed random variables. In this paper, we
compare the tail distribution of llSkll with that of llSjll, and deduce
some tail distribution maximal inequalities.
The main result of this paper* was inspired by the inequality from [I] that
says that
whenever XI and X, are independent identically distributed. Similar results for
L, ( p 3 1) such as ([XI1, g JIX,+X,Jj, are st~aightforward,at least if X, has
zero expectation. This inequality is-alsoobvious if either XIis symmetric or XI
is real valued positive. However, for arbitrary random variables, this result is
somewhat surprising to the author. Note that the identically distributed
assumption cannot be dropped, as one could take X, = 1 and X, = - 1.
In this paper, we prove a generalization to sums of arbitrarily many
independent identically distributed random variables. Note that all results in
this paper are true for Banach space valued random variables. The author
would like to thank Victor de la Peiia for helpful conversations.
THEOREM
1 . There exist universal constants c , = 3 and c, = 10 such thnt
i f X , , X , , ... are independent identically distributed random variables, and if we
k
set S, = C i = , X i , then for 1 <j < k
Pr(llsjil > t) G
Pr(llsklI '/cz).
This result cannot be asymptotically improved. Consider, for example, the
case where X I = 1 with very small probability, and XI = 0 otherwise. This
shows that for the inequality to be true for all X, the constant c, must be larger
than some universal constant for all j and k. Also, it is easy to see that c , must
*
Research supported in part by N.S.F. Grant DMS 9201357.
282
5. J. M o n t g o m e r y - S m i t h
be larger than some universal constant because it is easy to select X, and t so
that Pr(lJS,JJ> f) is dose to 1.
However, Latda [3] has been able to obtain the same theorem with
c , = 4 and c, = 5, or c, = 2 and c, = 7. In the case j = I, he has shown that
and these constants cannot be improved.
In oder to show this result, we will use the following definition. We will say
that x is a t-concentration point for a random variable X if Pr(1lX-xll< tj
> 2/3.
LEMMA
2. If x is a t-concentration point for X, and y is a t-concentration
and z is a t-concentration point for X+ I: then
point for
Ilx+y-zll
< 3t.
Proof. We have
Hence Pr(((x+y-zlj S 3t) > 0. Since x, y and z are fixed vectors, the result
follows. s
COROLLARY
3. If XI,X2,. .. are independent identically distributed random
variables, and if the partial sums S j =
Xi have i-concentration points sjfor
1 < j < k, then
xi=,
Proof. We prove the result by induction. It is obvious if j
Otherwise,
= k.
(The observant reader will notice that we are, in fact, following the steps
of Euclidean algorithm. The same proof could show that Ilksj-js,ll
d 3 V+ k-26a)t, where h is the greatest common divisor of j and k.) ta
,
P r o o f of T h e o r e m 1. We consider three cases. First suppose that
Pr (jS,-jll > 9t/10) d 1/3.
I
Note that 3,-Sj is independent of Sj, and identicaIIy distributed to Sk-j. Then
Pr(llSjll > t) d (3/2)Pr(l/Sj((> t and ]IS,-Sjl/
G (3/2)Pr(IlS,Jl > t/10).
< 9t/lO)
283
Sums of i.i.d, random vectors
For the second case, suppose that there is an i (1 C i G k) such that Si does
not have any (t/lO)-concentration point. Then
P ~ ( ( / S , + X , , ~...
+ +X,l( > t/lOla(X,+,, ..., XJ) 2 1/3,
and hence Pr (l(Sk(l> t/lO) 3 113 2 (113) Pr lllSill 3 t).
Finally, we are left with the third case where Pr(llSh-jI > 9t/10) > 1/3,
and Si has a (t/lO)-concentration point s, for afl 1 < i < k. clearly, llsk-jll
g 8t/10. Also, by Corollary 3,
Therefore,
Pr (llS,\l 2 t/lO) > Pr ([IS,-s,(l
and we are done.
< t/lO) 8 2/3 2 (2/3) Pr(llSjll > t ) ,
I
One might be emboldened to conjecture the following. Suppose that
X I , X,, ... are independent identically distributed random variables, and that
ui> 0. Let
k
S,
=
C criX,.
i= 1
Then one might conjecture that there is a universal constant such that for
ldjdk
Pr(llSi(l3 t) d c Fr(((S,II > t/c).
It turns out that this is not the case. Let Y,, Y,, ... be real valued
independent identically distributed random variables such that
Then, by the central limit theorem, there exists M 2 N~ such that
Now let Xi =
q+ l/M1/3, and
let
1
M
Then Pr (ISMI> 1/2) 2 1- 1/N, whereas Pr (lS,+X,+ , I > 3/N) < 2 / N .
We can obtain several corollaries to Theorem 1.
COROLLARY
4. There is a universal constant c such that if XI, X,,... are
independent identicauy distributed random variables, and if we set S, =
Xi,
then
Pr( sup (/Sjll> t) < cPr(/lSklI>t/c).
z=,
l<j<k
284
S. J. Montgomery-Smith
Latala [3] has been able to obtain this result with c,
with c , = 2 and c , = 8.
=
4 and c, = 6, or
P r o o f . This follows from Proposition 1.1.1 d [2J that states that if
XI,X,, . . . are independent (not necessarily identicaIIy dstributed), and if
Sk = C ki = l X i , then
Fr ( sup JISj(/> t ) < 3 sup Pr (llSjll > t/3).
1<j<k
l<j<k
It is also possible to prove this result directly using the techniques of the proof
of Theorem 1. The third case only requires that Pr(I(Skpj(J
> 9t/IO) > 1/3 for
one of j = I , 2, .. . , k, Hence, for the first case we may assume that
Pr(llS, - Sill > 9t/10) < 113 for all 1 g j < k. Let Aj be the event {JJSijl< t for
all i c j and I/Sj((> t). Then
Pr (Aj)< (3/2) Pr ( A j and 11 S, -Sjli
< 9JlO) 6 (312)Pr(Aj and
Summing over j, the result foIIows.
IISjII > t/loI.
ta
COROLLARY
5. There is a universal constant c such thut if X I , X,, ... are
independent identically distributed random variables, and if ifcti( < 1, then
P r o o f . The technique used in this proof is we11 known (see [2],
Proposition 1.2.2), but is included for completeness.
By taking real and imaginary parts of mi, we may suppose that the ai are
real. Without loss of generality, 1 $ a , 3 . . . 2 a, $ - 1. Then we may write
k
a..I= - 1
oi, where oi 2 0 . Thus
leil < 2, and hence
x.,
Applying Corollary 4, we obtain the result.
COROLLARY6. There are universal constants c , and c , such that if
X I , X, . .. are independent identically distributed random variables, and ifwe set
Sk= x i = , X i , then for 1 < k d j
Pr(lIsjll > t) d ( c l j / k )Pr(llSkl/ > kt/c,j).
P r o o f . Let m be the least integer such that rnk 2 j. By Theorem 1, it
follows that
The relation Pr (ljS,,(J > t) 6 m Pr (]ISk1 > t/m) is straightforward.
Sums of i.i.d. random uectors
285
The example where XIis constant shows that c, cannot be made smaller
than some universal constant. The example where XI = 1 with very small
probability and XI= 0 otherwise shows the same is true for c , .
REFERENCES
[I] V. H. do la Peiia and S. J. M o n t g o m e r y - S m i t h , Bounds on the tail probability or
U-statistics and quadratic forms, preprint.
[Z] S. Kwapietl and W. A. W o y c z y n s k i , Random Series and Stochastic Integrals: Simple and
Muhiple, Birkhauser, New York 1992.
[3] R.Lataka, A maxrrnaI mequality for sums of independent, identically distributed random cectors,
preprint, Warsaw Un~versity,Warszawa 1993.
Department of Mathematics
University of Missouri
Columbia, MO 65211, U.S.A.
Received un 20.12.1993
9
-
PAMS 14.2